Given \(M \in D^b_{cons}(Y_{proet},\mathbf {Z}_\ell )\) , we must show that each \(\mathcal {H}^i(M)\)  lies in \(D^b_{cons}(Y_{proet},\mathbf {Z}_\ell )\) . Using the definition of constructibility [1]}, we must show that the abelian pro-étale sheaves \(\mathcal {H}^i(M)/\ell \)  and \(\mathcal {H}^i(M)[\ell ]\)  are constructible \(\mathbf {F}_\ell \) -sheaves for all \(i\) . Note that these sheaves can be regarded as subobjects (resp. quotient objects) of some \(\mathcal {H}^i(M/\ell )\)  via the Bockstein sequence for \(\ell \) . As étale subquotients of étale constructible constructible sheaves on a noetherian scheme are constructible [2]}, it suffices to show that the pro-étale sheaves \(\mathcal {H}^i(M)/\ell \)  and \(\mathcal {H}^i(M)[\ell ]\)  are classical. In fact, by the Bockstein sequence and stability of classical sheaves under cokernels in all pro-étale sheaves, it suffices to prove that each \(\mathcal {H}^i(M)/\ell \)  is étale. By definition of constructibility, we know that \(\mathcal {H}^i(M/\ell ^n)\)  is classical for all \(i\)  and \(n\) . As classical sheaves are stable under images, it is then enough to show that \(\mathcal {H}^i(M)/\ell \subset \mathcal {H}^i(M/\ell )\)  is exactly the image of \(\mathcal {H}^i(M/\ell ^n) \rightarrow \mathcal {H}^i(M/\ell )\)  for \(n \gg 0\) . By the Bockstein sequences for \(\ell ^n\) , this would follow if we knew that the projective system \(\lbrace \mathcal {H}^i(M)[\ell ^n]\rbrace _{n \ge 1}\)  are Mittag-Leffler for each \(i\) , i.e., if each \(\mathcal {H}^i(M)\)  had bounded \(\ell \) -power torsion. If \(M\)  is lisse, this is clear. In general, recall the following fact from [1]}: if \(k:Z \rightarrow Y\)  is a (necessarily constructible, as \(Y\)  is noetherian) locally closed immersion, then the functors \(k^*\)  and \(k_!\)  on the derived category of all pro-étale sheaves preserve limits and colimits and commute with \(\mathcal {H}^i(-)\) . By [1]}, we know that \(Y\)  admits a finite stratification \(\lbrace k_j:Y_j \rightarrow Y\rbrace \)  such that each \(N_j := k_j^* M\)  is lisse. The aforementioned properties of \(k_{j,!}\)  and \(k_j^*\)  then show that \(\mathcal {H}^i(M)[\ell ^n]\)  admits a finite filtration whose graded pieces have the form \(k_{j,!} k_j^* (\mathcal {H}^i(N_j)[\ell ^n])\)  for lisse complexes \(N_i\) . But then each \(\mathcal {H}^i(N_j)\)  is also lisse and hence has bounded \(\ell \) -power torsion, so the corresponding claim for \(\mathcal {H}^i(M)\)  follows by devissage.
